Revd Jim Percival and his daughter Ruth were originally arrested by police in 2014 on suspicion and conspiracy to conceal the birth of a child, after police found the body of a baby boy at their home in Freckleton, Lancashire.
After investigating police took no further action regarding those offences and released the pair.
It is still not clear how the baby died.
Mr Percival, formerly of Holy Trinity Church Freckleton (above), and his daughter were then arrested again last Thursday.
In a statement, a police spokesman said: "They were interviewed at a police station in the county during the course of Thursday and have been released on bail until February 16 pending further enquiries.
"Both the man and the woman had been previously arrested on suspicion of murder and conspiracy to conceal the birth of a child. Both have now been released no further action in respect of those offences.
"Police were called to an address in Freckleton at shortly before 6pm on Tuesday, November 25 2014 following an initial report that a woman had given birth to a still-born baby.
"An investigation has been launched and a post mortem examination has been carried out."